Geometric Modeling - Parametric Representation of Synthetic Curves

Сентябрь 13, 2022

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Geometric Modeling - Parametric Representation of Synthetic Curves

Содержание

2. * Planar vs. Space
3. * Analytic (known form) vs. Synthetic (free form) Creating these curves by using known analytic curve
4. * Interpolation vs. Approximation The curve passing through given data (control) points - interpolation curve. The
5. * Continuity The smoothness of the connection of two curves or surfaces at the connection points
6. * Cubic Curves In an expanded vector form: Parametric equation of a cubic spline segment: where
7. * Hermite Cubic Splines Hermite form of a general cubic spline is defined by positions and
8. * Hermite Cubic Splines
9. * Hermite Cubic Spline – Tangent Vector
10. * X Hermite Cubic Splines - example The Hermite curve fits the points: P0 = [1,1]T,
11. * Bezier Curves - sl. 1 Parametric equation of Bezier curve where P(u) is the position
12. * Bezier Curves - sl. 3 For n = 3: Bezier basis matrix MB Or, in
13. * Bezier Curves - sl. 2 General Characteristics The Bezier curve is defined by n+1 points
14. * Bezier Curves - sl. 5 Practice The coordinates of 4 control points are given: P0
15. * Bezier Curves - sl. 4
16. * B-spline Curves - sl. 1 See: http://www.ibiblio.org/e-notes/Splines/Basis.htm Powerful generalization of Bezier curves local control opportunity
17. * B-spline Curves - sl. 2 See: http://www.ibiblio.org/e-notes/Splines/Basis.htm The B-spline curve defined by n+1 control points
18. * B-spline Curves - sl. 3 Basis Functions The function Ni,k determines how strongly control point
19. * NURBS Curves - sl. 1 NURBS (Non-uniform Rational B-spline) curves are the generalization of uniform
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Planar vs. Space

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Analytic (known form) vs. Synthetic (free form)
Creating these curves by using

known analytic curve equations is not reasonable all the time. Sometimes – impossible.

We can create simplistic objects such as the forklift given below by using known equations.

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Interpolation vs. Approximation
The curve passing through given data (control) points

- interpolation curve.
The curve not necessarily passing but controlled by data points - approximation curve

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Continuity
The smoothness of the connection of two curves or surfaces at

the connection points or edges.

C0: simple connection of two curves
C1: the geometric slopes at the joint must be same
C2: curvature continuity that not only the gradients but also the center of curvature is the same

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Cubic Curves
In an expanded vector form:
Parametric equation of a cubic spline

segment:
where 0u 1

The tangent vector:

In an expanded vector form:

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Hermite Cubic Splines
Hermite form of a general cubic spline is defined

by positions and tangent vectors at two data points.

Charles Hermite
(1822 - 1901)

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Hermite Cubic Splines

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Hermite Cubic Spline – Tangent Vector

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X
Hermite Cubic Splines - example
The Hermite curve fits the points:
P0

= [1,1]T,
P1 = [3,5]T
and the tangent vectors: P0’ = [0,4]T,
P1’ = [4,0]T.
Calculate
the parametric mid-point of the curve,
the tangent vector on that point.
Sketch the curve on the grid

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Bezier Curves - sl. 1
Parametric equation of Bezier curve
where P(u)

is the position vector of a point on the curve, Pi are control points, and Bi,n are the Bernstein polynomials (blending functions for the curve).
and C(n,i) are the binomial coefficients:
In an expanded form:

Pierre Bezier
(1910-1999)
Renault

Paul de Casteljau
(1930)
Citroën

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Bezier Curves - sl. 3
For n = 3:
Bezier basis matrix
MB
Or, in

matrix form:

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Bezier Curves - sl. 2
General Characteristics
The Bezier curve is defined by

n+1 points
Only P0 and Pn+1 lie on the curve
The curve is tangent to the first and last polygon segments
The curve shape tends to follow the polygon shape.
Convex hull property.
The sum of Bi,n functions is always equal to unity.

Bezier vs. Hermite Cubic Spline
The Bezier curve is controlled by data points. No derivatives
The order is variable: n+1 points define nth order curve . -> higher order continuity

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Bezier Curves - sl. 5 Practice
The coordinates of 4 control points are

given:
P0 = [2,2]T, P1 = [2,3]T, P3 = [3,3]T, P4 = [3,2]T
Find the equation of the resulting Bezier curve,
Find the points on the curve for u = 0, ¼, ½, ¾, 1,
Sketch the curve.

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Bezier Curves - sl. 4

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B-spline Curves - sl. 1 See: http://www.ibiblio.org/e-notes/Splines/Basis.htm
Powerful generalization of Bezier curves
local

control
opportunity to add control points without increasing the degree of the curve
ability to interpolate or approximate data points
The B-spline curve defined by n+1 control points Pi consists of n – 2 curve segments and is given by:
where Ni,k(u) are the B-spline (blending or basis) functions. The parameter k controls the degree (k-1) of the B-spline curve.

Local control

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B-spline Curves - sl. 2 See: http://www.ibiblio.org/e-notes/Splines/Basis.htm
The B-spline curve defined by

n+1 control points Pi consists of n – 2 curve segments and is given by:
where Ni,k(u) are the B-spline (blending or basis) functions. The parameter k controls the degree (k-1) of the B-spline curve.

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